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Spectra of tensor triangulated categories over category algebras. (English) Zbl 1309.18014

An EI category is a category such that any endomorphism is an isomorphism. Typical examples include groups (seen as categories with one object) and posets. One may define a tensor product on representations of an EI category by taking a pointwise tensor product. This tensor product is exact, and thus turns the bounded derived category into a tensor triangulated category.
For tensor triangulated categories, P. Balmer [J. Reine Angew. Math. 588, 149–168 (2005; Zbl 1080.18007)] introduced a spectrum (often called the Balmer spectrum) which is to be thought of as an analog of the spectrum of a ring in commutative algebra. In particular Balmer’s spectrum can be used to classify tensor ideals.
In the paper under review, the author studies the Balmer spectrum of the bounded derived category of the category algebra \(k \mathcal{C}\) of a finite EI category \(\mathcal{C}\). It is shown that, as a topological space, this spectrum is the union of the spectra of the group algebras of all endomorphism groups of objects in \(\mathcal{C}\). These in turn can be understood as homogeneous spectra of the corresponding group cohomology rings.
The author focuses in particular on the case that \(\mathcal{C} = G \propto \mathcal{P}\) is a semidirect product of a group and a finite poset. In this case the category algebra is Gorenstein, and one obtains an induced tensor product on the stable category of Cohen-Macaulay modules. It is shown that the (underlying topological space of the) Balmer spectrum of this tensor triangulated category can again be understood as a union, this time of the projective spectra of group cohomology rings.

MSC:

18E30 Derived categories, triangulated categories (MSC2010)
16S35 Twisted and skew group rings, crossed products
16E35 Derived categories and associative algebras

Citations:

Zbl 1080.18007
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References:

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